Maximum Split of Positive Even Integers

Maximum Split of Positive Even Integers

1. What is this problem about?

The Maximum Split of Positive Even Integers coding problem asks you to take a given positive even integer finalSum and split it into the maximum number of unique positive even integers. For example, if finalSum is 12, you could split it into [2, 10] or [2, 4, 6]. The split [2, 4, 6] is better because it has 3 unique integers.

2. Why is this asked in interviews?

Google uses this problem to test a candidate's mathematical intuition and greedy logic. It requires you to realize that to maximize the count of unique numbers, you should pick the smallest possible unique even integers (2, 4, 6, 8...) until you can no longer add a unique even number without exceeding the sum.

3. Algorithmic pattern used

This utilizes the Math and Greedy interview pattern. Start with the smallest positive even integer, 2. Subtract it from the finalSum and add it to your list. Then try 4, then 6, and so on. In each step, if the remaining finalSum is less than or equal to the next even number you would pick, you simply add the entire remaining finalSum to the last number in your list to ensure all numbers remain unique and even.

4. Example explanation

finalSum = 14.

1. Pick 2. Remaining = 12. List: [2].
2. Pick 4. Remaining = 8. List: [2, 4].
3. Next would be 6. But 8 is only slightly larger than 6. If we picked 6, we'd have 2 left. But we already used 2!
4. Instead, add the remaining 8 to the last element: [2, 4 + 8] = [2, 12]. Wait, a better split: 2 + 4 + 8. Is there another? 2 + 4 + ... 14 - 2 - 4 = 8. Since 8 is not in our list {2, 4}, we can actually just pick 8. Revised: [2, 4, 8]. Sum is 14. All unique and even. Count is 3.

5. Common mistakes candidates make

A common mistake is trying to use backtracking to find all possible splits, which is unnecessary and too slow. Another error is not handling the "odd" case, although the problem usually specifies that finalSum is even. Candidates also sometimes forget to check if the last remaining chunk is already in their list, which is why the strategy of adding it to the previous element is so robust.

6. Interview preparation tip

When asked to "maximize the number of items" in a sum, the greedy choice is almost always to pick the smallest available items. This is a recurring theme in "Greedy, Math interview pattern" questions.